Algebraswithternary lawofcompositionandtheir ...

[Pages:17]Journal of Generalized Lie Theory and Applications

Vol. * (20**), No. *, 1???

arXiv:0901.2506v1 [math.RA] 16 Jan 2009

Algebras with ternary law of composition and their

realization by cubic matrices

V. Abramov a, R. Kerner b, O. Liivapuu a and S. Shitov a

a Institute of Mathematics, University of Tartu, Liivi 2, Tartu 50409, Estonia E-mail: viktor.abramov@ut.ee, olgai@ut.ee, sergei.shitov@ut.ee

b LPTMC, Tour 24, Bo^ite 121, 4, Place Jussieu, 75252 Paris Cedex 05, France E-mail: rk@ccr.jussieu.fr

Abstract

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions we construct a ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.

2000 MSC: 17A40, 20N10.

1 Introduction

A ternary algebra or triple system is a vector space A endowed with a ternary law of composition : A ? A ? A A which is a linear mapping with respect to each its argument, and we will call this mapping a ternary multiplication or triple product of a ternary algebra A. Hence a ternary algebra is an algebra which closes under a suitable triple product. Obviously any binary algebra which closes under double product can be considered as a ternary algebra if one defines the ternary multiplication as twice successively applied binary one, and in this case the ternary multiplication is generated by a binary one. However there are ternary multiplications which can not be obtained as twice successively applied binary multiplication. For instance, pure imaginary numbers or elements of grading one of a superalgebra closes under triple product. A well known example of a ternary matrix algebra is the vector space Matm,n of m ? n matrices endowed with the ternary multiplication (A, B, C) = A ? BT ? C, where A, B, C Matm,n and BT is transpose of the matrix B. Since Lie algebras play a fundamental role in physics, particular attention was given to ternary algebras when they were shown to be building blocks of ordinary Lie algebras. Given ternary algebra one can construct a Lie algebra by using the method proposed by Kantor in [10]. This method was extended to super Lie algebras in [7] and later was applied by the same authors in [8] to construct a gauge field theory by introducing fundamental fields associated with the elements of a ternary algebra.

A skew-symmetric bilinear form is an important component in the large class of algebraic structures such as Lie algebras, Grassmann algebras and Clifford algebras. For example, the Lie brackets [ , ] : L ? L L of a Lie algebra L is the skew-symmetric bilinear form, and the multiplication (a, b) G ? G a ? b G of a Grassmann algebra G restricted to the

2

V. Abramov, R. Kerner, O. Liivapuu and S. Shitov

subspace of odd elements is the skew-symmetric bilinear form. A skew-symmetry of a bilinear form can be interpreted by means of the faithful representation of the symmetric group S2 = {e, } {1, -1}, where e is the identity permutation, as follows: a bilinear form ? is skewsymmetric if ?(x(1), x(2)) = (-1) ?(x1, x2). Making use of this interpretation we can construct a ternary analog of a skew-symmetric bilinear form replacing S2 by Z3 S3 with its faithful representation by cubic roots of unity j = e2i/3, i.e. Z3 = {e, 1, 2} {1, j, j2}, where e is the identity permutation and 1, 2 are the cyclic permutations, as follows: a trilinear form is called j-skew-symmetric if for any elements a, b, c of a vector space A it satisfies

(a, b, c) = j (b, c, a) = j2 (c, a, b).

The notion of a j-skew-symmetric form can be assumed as a basis for a ternary analog of Grassmann, Clifford and Lie algebras. These ternary structures were developed in [1, 2, 13, 15] and applied to construct a ternary analog of supersymmetry algebra in [3, 11, 12, 14].

In this paper we study algebras with ternary law of composition. In Section 2 we consider partially and totally associative ternary algebras of first and second kind. We show that a triple product of a ternary algebra induces three binary multiplications and find the sufficient and necessary condition a triple product of a ternary algebra must satisfy in order to induce the associative binary algebra. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. The sufficient and necessary condition a ternary multiplication must satisfy in order to induce a structure of associative binary algebra in each fiber is given in terms of the vector bundle over a ternary algebra. The relations for different kinds of partial and total associativity of a ternary algebra and induced by it binary algebras are found in the terms of the structure constants of a ternary algebra. It should be pointed out that the cohomologies of a ternary algebra of associative type are studied in [5].

In Section 3 we consider an algebraic structure consisting of two bimodules over unital associative algebras with involution and construct a ternary algebra by means of this algebraic structure. Choosing different modules, unital associative algebras and homomorphisms we show that this structure allows to construct a large class of ternary algebras including a ternary algebra of rectangular matrices and ternary algebras of sections of a vector bundle over a smooth finite dimensional manifold. We end the Section 3 by constructing the binary Lie algebra of matrices whose entries are the elements of bimodules and unital associative algebras. It should be mentioned that there are n-ary generalizations of Lie algebra which include the concepts such as n-ary algebra of Lie type enclosing n-ary Nambu algebra, n-ary Nambu-Lie algebra. The concept of n-ary Hom-algebra structure generalizing previously mentioned n-ary generalizations of Lie algebra is introduced and studied in [6]. A good and detailed survey on the theory of ternary algebras can be found in [4].

It is well known that a large class of associative algebras can be constructed by means of square matrices and their multiplication. Though the rectangular matrices can be successfully used to construct a ternary algebra we think that probably more appropriate objects to construct ternary algebras are the cubic matrices. Our aim in Section 4 is to construct ternary algebras of cubic matrices and to study their structures. We find four different totally associative ternary multiplications of second kind of cubic matrices and prove that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. It is worth mentioning that our search for associative ternary multiplications of cubic matrices has shown that there is no totally associative ternary multiplication of first kind in the case of cubic matrices. I Section 5 we describe the ternary analog of Lie algebra of cubic matrices of second order by finding all commutation relations of generators of this algebra with respect to j-commutator.

Algebras with ternary law of composition and their realization by cubic matrices

3

2 Algebras with ternary law of composition

In this section we remind a notion of a ternary algebra and its partial or total associativity of first or second kind. Holding fixed one argument of a ternary multiplication we get the binary multiplications and study the relation between the associativity of a ternary multiplication and the associativity of induced binary multiplication. We propose to use a vector bundle approach to describe the family of binary algebras induced by a ternary algebra.

Let A, B be complex vector spaces, and : A ? A ? A B be a B-valued trilinear form. We will call a ternary law of composition or ternary multiplication on A if is a A-valued trilinear form. The pair (A, ) is said to be a ternary algebra or triple system if A is a complex vector space, and : A ? A ? A A is a ternary law of composition on A. It is obvious that relation analogous to binary associativity in the case of ternary law of composition should contain at least five elements of A. There are three different ways to apply twice a ternary multiplication to ordered sequence of five elements a, b, c, d, f A which lead us to the following relations defining a notion of partial associativity for ternary multiplication:

( (a, b, c), d, f ) = (a, b, (c, d, f )),

(1)

( (a, b, c), d, f ) = (a, (b, c, d), f ),

(2)

(a, (b, c, d), f ) = (a, b, (c, d, f )).

(3)

Hence we have three different kinds of partially associative ternary algebra (A, ) which will be called lr-partially associative ternary algebra (1), lc-partially associative ternary algebra of first kind (2) and cr-partially associative ternary algebra of first kind (3). A ternary algebra (A, ) is said to be totally associative ternary algebra of first kind if its ternary multiplication satisfies any two of the relations (1)?(3). It is obvious that in the case of totally associative ternary algebra of first kind a ternary multiplication satisfies the relations

( (a, b, c), d, f ) = (a, (b, c, d), f ) = (a, b, (c, d, f )),

(4)

where a, b, c, d, f A. The notion of totally associative ternary algebra of first kind can be viewed as a direct ternary generalization of classical associativity ?(?(x, y), z) = ?(x, ?(y, z)), where x, y, z are the elements of an algebra (A, ?) with a binary law of composition ? : A ? A A, when one applies twice algebra multiplication (binary or ternary) to ordered sequence of elements of algebra successively shifting the first (interior) multiplication from left to right and setting equal obtained products. In this sense the notion of lr-partial associativity can be considered as most similar to classical associativity whereas the notion of lc-partial or cr-partial associativity can be defined for the first time only in the case of ternary multiplication because in the case of binary multiplication there is no central group of two elements in the middle of a sequence x, y, z A.

Since the notion of lc-partial or cr-partial associativity appears for the first time in the case of ternary multiplication there is no reason to keep the requirement of fixed order of a sequence a, b, c, d, f A looking for a possible analogs of associativity in the case of ternary algebras. It turns out that we get a useful notion of ternary associativity giving up the requirement of fixed order of elements in a sequence a, b, c, d, f A. This means that unlike the case of ternary associativity of first kind we not only successively shift the first (interior) multiplication inside a sequence of elements a, b, c, d, f A from left to right but at the same time permute the elements b, c, d in the middle of sequence. Obviously we should use non-cyclic permutation in order to get the initial order of a sequence a, b, c, d, f on the second step. This reasoning leads us to the following relations:

( (a, b, c), d, f ) = (a, (d, c, b), f ),

(5)

(a, (d, c, b), f ) = (a, b, (c, d, f )).

(6)

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V. Abramov, R. Kerner, O. Liivapuu and S. Shitov

A ternary algebra (A, ) is said to be lc-partially associative ternary algebra of second kind if ternary multiplication satisfies (5) and cr-partially associative ternary algebra of second kind if satisfies (6). A ternary algebra (A, ) is said to be totally associative ternary algebra of second kind if it is lr-partially associative and either lc-partially associative of second kind or cr-partially associative of second kind. Hence in the case of totally associative ternary algebra of second kind we have

( (a, b, c), d, f ) = (a, (d, c, b), f ) = (a, b, (c, d, f )).

(7)

A ternary multiplication of ternary algebra (A, ) has three arguments (a, b, c), where a, b, c A, and if we fix one of them then induces the binary multiplication on A. It is obvious that this allows us to split the ternary multiplication into three binary ones. We can study the structure of ternary multiplication from this point of view by making use of known concepts and methods of the theory of binary algebras. Given an element a A a ternary multiplication induces three binary multiplications a1, a2, a3 on A defined as follows:

a1(b, c) = (a, b, c), a2(b, c) = (b, a, c), a3(b, c) = (b, c, a),

(8)

where b, c A. The binary multiplications a1, b2, c3 are not independent because of the relations

a1(b, c) = b2(a, c) = c3(a, b).

(9)

A vector space A equipped with the binary multiplication ai, i = 1, 2, 3 becomes the binary algebra which will be denoted by (A, ai). Considering an element a in (A, ai) as a parameter ranging within a vector space A we have three families of binary algebras (A, a1), (A, a2), (A, a3) induced by a ternary multiplication . The family of binary algebras (A, ai) is said to be an associative family of binary algebras induced by a ternary algebra (A, ) if for any a, b, c, d, f A

it holds

ai(bi(c, d), f ) = bi(c, ai(d, f )).

(10)

Taking a = b in the previous relation we see that each associative family of binary algebras (A, ai ) is the family of associative binary algebras.

It is useful to describe the above mentioned families of binary algebras in terms of vector

bundle. For this purpose we will assume that A is a topological vector space, and a ternary

multiplication : A ? A ? A A is a continuous mapping. Let us consider the direct product

E = A ? A as the trivial vector bundle over the base space A with the fiber A and the projection : E A defined as usual (p) = a, where p = (a, b) E. Any fiber -1(a) of E is

isomorphic to A, and we will denote this isomorphism at a point a of the base space A by a, i.e. a : -1(a) A and a(p) = b, where p = (a, b) -1(a). Let a, b A be two points of the base space of a vector bundle E. Then

ab = -b 1 a : -1(a) -1(b),

(11)

is the isomorphism between two fibers.

In order to apply the constructed vector bundle E to describe the families of binary algebras

induced by a ternary algebra (A, ) within the framework of a single structure we assume that

the base space A of this bundle is equipped with a ternary multiplication . For any point a A of the base space a fiber -1(a) E at this point is endowed with one of the binary multiplications a1, a2, a3 which we carry over from the family of binary algebras to fibers of E by requiring a to be an isomorphism of algebras, i.e.

a(ai(p, q)) = ai(a(p), a(q)),

(12)

Algebras with ternary law of composition and their realization by cubic matrices

5

where p, q -1(a). If each fiber of E is endowed with a binary multiplication i then in order

to emphasize this algebraic structure of fibers we will denote the corresponding vector bundle by Ei. Thus Ei = (A, ) ? (A, i), where the base space (the first factor in the direct product) is a ternary algebra (A, ), and a fiber -1(a) is the binary algebra (A, ai ). We will call Ei, where i = 1, 2, 3, the vector bundle of binary algebras over a ternary algebra (A, ). A section of the

vector bundle Ei is a continuous mapping : A Ei satisfying = idA, and the vector space of continuous sections will be denoted by (Ei). Evidently this vector space equipped with the binary multiplication

i(, ) (a) = ai (a), (a) ,

(13)

where , (Ei), is the binary algebra. The notion of an associative family of binary algebras defined by (10) can be described in

the terms of vector bundle Ei. Let p, q Ei be two points of a vector bundle Ei and a A. A vector bundle of binary algebras Ei is said to be an associative vector bundle of binary algebras over a ternary algebra (A, ) if for any p, q Ei and a A it holds

(q) i (q) ((pq)) i (p)(p, -(1p)(a)) , q = (p) i (p) p, ((pq)) i (q)(-(1q)(a), q) . (14)

Particularly (14) implies the associativity of a fiber -1(a) for any a A if we take (p) = (q)

in (14), i.e. any associative vector bundle of binary algebras Ei is a vector bundle of associative binary algebras whereas the converse is generally not true. Now it is natural to pose a question concerning the associativity of induced binary algebras (A, ai) provided a ternary algebra (A, ) is partially or totally associative of first or second kind.

Proposition 1. A ternary algebra (A, ) is lr-partially associative ternary algebra if and only if E2 is the associative vector bundle of binary algebras over a ternary algebra (A, ). Particularly if a base space (A, ) is lr-partially associative ternary algebra then each fiber of the vector bundle E2 is an associative binary algebra and the binary algebra of sections of this bundle (E2) is associative algebra.

Indeed the left side of (14) can be transformed as follows:

(q) 2(q) ((pq)) 2(p)(p, -(1p)(a)) , q = (q) 2(q) 2(p) -(1q) (p)(p), -(1q)(a) , q = 2(q) 2(p) (p)(p), a , (q)(q) = (p)(p), (p), a , (q), (q)(q) .

Analogously for the right side of (14) we have

(p) i (p) p, ((pq)) i (q)(-(1q)(a), q) = (p)(p), (p), a, (q), (q)(q) ,

and this proves the lr-partial associativity of a ternary algebra (A, ). It is well known that any associative binary algebra is the Lie algebra with respect to the

commutator defined with the help of a binary multiplication of this algebra, and the associativity of a binary multiplication implies the Jacoby identity for the commutator. It follows from the Proposition 1 that if a ternary algebra (A, ) is lr-partially associative ternary algebra then each fiber -1(a), where a A, of the vector bundle E2 is the Lie algebra with respect to the commutator [ , ]a defined by

[p, q]a = a2(p, q) - a2(q, p),

(15)

where p, q -1(a). Clearly the associative binary algebra of sections (E2) is the Lie algebra under the commutator

[, ](a) = [(a), (a)]a, where , (E2).

(16)

6

V. Abramov, R. Kerner, O. Liivapuu and S. Shitov

A ternary algebra (A, ) is said to be a ternary algebra of Lie type of first kind (of second kind) if for any a1, a2, a3 A it holds

(a(1), a(2), a(3)) = 0

S3

(a(1), a(2), a(3)) = 0 ,

Z3

(17)

where S3 is the symmetry group of third order and Z3 is its cyclic subgroup. Clearly any ternary algebra of Lie type of second kind is a ternary algebra of Lie type of first kind whereas

the converse is generally not true. From (17) it follows that any element a of a ternary algebra of Lie type (of first or second order) satisfies a3 = (a, a, a) = 0. It is pointed out in the

Introduction that we can construct a ternary analog of the notion of skew-symmetry by means

2i

of a faithful representation of Z3 by cubic roots of unity. Let j = e 3 C be the primitive cubic root of unity. A ternary multiplication of a ternary algebra (A, ) is said to be j-skew-

symmetric if

(a, b, c) = j (b, c, a) = j2 (c, a, b),

(18)

where a, b, c A. If a ternary multiplication of (A, ) is j-skew-symmetric then (A, ) is a ternary algebra of Lie type of second order. Indeed in this case we have

(a, b, c) + (b, c, a) + (c, a, b) = (a, b, c) + j2 (a, b, c) + j (a, b, c) = 0.

(19)

We see that the notion of j-skew-symmetric ternary multiplication is based on the faithful representation of the cyclic group Z3 by cubic roots of unity. Given a ternary algebra (A, ) we can make it the ternary algebra of Lie type of second order by endowing it with the ternary j-brackets or the ternary j-commutator which is defined by

[a, b, c] = (a, b, c) + j (b, c, a) + j2 (c, a, b).

(20)

If A has an involution : A A then will be called Hermitian if it satisfies (a, b, c) = (c, b, a). Hence a Hermitian j-skew-symmetric ternary multiplication satisfies (18) and

(a, b, c) = (c, b, a) = j2 (b, a, c) = j (a, c, b).

(21)

Let us suppose that (A, ) is a lr-partially nonassociative ternary algebra, i.e. in general we have ((a, b, c), d, f ) = (a, b, (c, d, f )), where a, b, c, d, f A. A ternary algebra (A, ) is said to be a ternary algebra of Jordan type if its ternary multiplication satisfies the following identities:

(a, b, c) = (c, b, a),

(22)

((a, b, c), b, (a, b, a)) = (a, b, (c, b, (a, b, a))),

(23)

where a, b, c A. It is easy to see that if (A, ) is a ternary algebra of Jordan type then for any a A the binary algebra (A, a2) is the Jordan algebra. Indeed in this case the identities (22,23) take on the form

a2(b, c) = a2(c, b), a2(a2(b, c), a2(b, b)) = a2(b, a2(c, a2(b, b))).

(24)

Proposition 2. If (A, ) is lr-partially associative ternary algebra then the ternary algebra (A, ), where (a, b, c) = (a, b, c) + (c, b, a) is the ternary algebra of Jordan type.

We see that having fixed one variable in a triple product (a, b, c) of a ternary algebra (A, ) we can study the structure of a ternary multiplication by splitting it into three binary ones. What kind of structures induces a ternary multiplication of (A, ) if one fixes two variables in

Algebras with ternary law of composition and their realization by cubic matrices

7

(a, b, c)? Obviously fixing two variables we get the linear operator acting on A, and this is the

second way for studying the structure of a ternary multiplication. Let Lin(A) be the algebra of linear operators of the vector space A. Given a pair (a, b) A ? A we define the linear operators Li(a, b) : A A, where i = 1, 2, 3, as follows:

L1(a, b) ? c = (c, a, b), L2(a, b) ? c = (a, c, b), L3(a, b) ? c = (a, b, c).

(25)

Actually these operators are not independent because for any a, b, c A we have the relations

L1(c, b) ? a = L2(a, b) ? c = L3(a, c) ? b.

(26)

It is easy to see that for every i the linear operator Li(a, b) is bilinear with respect to its variables a, b, and therefore the family of linear operators {Li(a, b)}(a,b)A?A determines the bilinear mapping Li : A ? A Lin(A). On the other hand if there is a vector space A equipped

with a bilinear mapping L : A ? A Lin(A) then one can construct the ternary algebra (A, )

by letting (a, b, c) = L(a, b) ? c.

Now we can introduce an analog of identity element for a ternary algebra (A, ) by means of the bilinear mappings Li : A ? A A, i = 1, 2, 3. Indeed given an element a A we define I(ai) A ? A by

I(ai) = {(e, e~) A ? A : L(i)(e, e~) = idA},

(27)

where idA Lin(A) is the identity operator. A pair (e, e~) A ? A is said to be an identity i-pair

for an element a if (e, e~) Ia(i). Let I(i) =

aA Ia(i) and I =

3 i=1

I(i)

.

If

I(i)

=

then we will

call an element (e, e~) I(i) an identity i-pair, and similarly if I = then we will call an element

(e, e~) I an identity pair of a ternary algebra (A, ).

Let us now assume that the vector space A of a ternary algebra (A, ) is a finite dimensional

vector space, i.e. A is an r-dimensional vector space and e = {e1, e2, . . . , er} is a basis for A. Then for any element a A we have a = ae, and the triple product of elements a, b, c A

can be expressed as follows:

(a, b, c) = (ae, b e, c e) = C abc e,

where C are the structure constants of a ternary algebra (A, ) defined by

(e, e , e ) = C e.

(28)

If e = {e1, e2 , . . . er} is another basis for a vector space A and e = Ae, where A = (A) is the transition matrix then

C = AAA A?C,

where C are the structure constants of a ternary algebra (A, ) with respect to a basis e, i.e. (e, e , ec) = C e = C A e, and A-1 = (A?) is the inverse matrix of A.

If we require a ternary algebra (A, ) to be a partially or totally associative ternary algebra

either of first or second kind then this requirement leads to the relations the structure constants

of (A, ) have to satisfy. These relations for different kinds of associativity of first kind have the

following form:

C C? = CC? C C? = C?C C?C = CC?

(lr-partial associativity of first kind), (lc-partial associativity of first kind), (cr-partial associativity of first kind).

8

V. Abramov, R. Kerner, O. Liivapuu and S. Shitov

It follows from the above relations that if (A, ) is a totally associative ternary algebra of first kind then the structure constants satisfy

C C? = C?C = CC?.

In the case of ternary associativity of second kind we have the following relations:

C C? = C?C (lc-partial associativity of second kind), C?C = CC? (cr-partial associativity of second kind).

The structure constants of a totally associative ternary algebra of second kind satisfy

C C? = C?C = CC?.

For the

any a binary

mAultaiptleicrantairoynsaldgeefibnraed(Aby, r)eliantdiouncses(8th).reTe hbeinsatrryucatlugreebrcaosns(tAa,ntais),Kii,=(a1),

2, 3, with of binary

algebra (A, ai) defined by ai(e, e) = Ki,(a)e can be expressed in terms of the structure

constants of a ternary algebra (A, ) as follows:

K1,(a) = C a, K2,(a) = C a , K3,(a) = C a , where a = a e.

(29)

3 Lie algebras from ternary algebra

In this section we propose few different methods for constructing ternary algebras and apply these methods to construct a ternary algebra of vector fields on a smooth finite dimensional manifold and a ternary algebra of rectangular matrices. Particularly the curvature of an affine connection determines the structure of a ternary algebra on the module of vector fields on a smooth manifold. Given two modules over the algebras with involutions we construct a ternary algebra which is used to construct a Lie algebra. Our approach generalizes the approach proposed in [7, 8], where the authors use the rectangular complex matrices.

Let A be a vector space over the complex numbers C and A be the dual space. Given a C-multilinear mapping T : A ? A ? A ? A C we construct the ternary algebra (A, ) by defining the ternary multiplication as follows:

( (a, b, c)) = T (a, b, c, ),

(30)

where a, b, c A, A. Particularly given a C-bilinear mapping L : A ? A Lin(A), where Lin(A) is the algebra of linear operators of a vector space A, we define

T (a, b, c, ) = (L(a, b) ? c),

(31)

and applying (30) we get the ternary algebra (A, ) whose ternary multiplication can be described implicitly by the formula

(a, b, c) = L(a, b) ? c, a, b, c A.

(32)

Applying this construction to a module over an associative unital algebra we can construct a ternary algebra by means of (30) or (32). Indeed if A is a left A-module, where A is a binary unital associative complex algebra, A is the dual module and T : A ? A ? A ? A A is an A-multilinear mapping then (A, ) is the ternary algebra with the ternary multiplication defined by (30). Similarly given an A-module A and a A-bilinear mapping L : A ? A Lin(A), where

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